Frequency and phase estimation for MPSK signals

ABSTRACT

A frequency and phase estimator simultaneously estimates the frequency and phase of an MPSK modulated signal with a frequency uncertainty range on the order of the symbol rate. The estimator defines a plurality of contiguous bands within the frequency uncertainty range of the signal, estimates the frequency to one of the bands, and utilizes the frequency estimate to derive a phase estimate. In a preferred embodiment, a plurality of signal samples of the frequency shifted signal in each of said bands are accumulated to produce a vector for each band, and the frequency estimate is selected in one of said bands, based upon the magnitude of the corresponding vector. The phase is estimated from the argument of the corresponding vector. The present invention is particularly suited for burst modems or TDMA systems, where frequency and phase estimates must be derived reliably from a limited number of incoming symbols at the beginning of each burst.

FIELD OF THE INVENTION

The present invention relates generally to data communications signalprocessing and, more specifically, concerns a frequency and phaseestimation method and apparatus for an MPSK modulated carrier.

BACKGROUND OF THE INVENTION

Many types of data communications systems transfer information (e.g.,audio or video signals) by modulating the information onto a carriersignal such as a sine wave. The carrier is modulated by varying one ormore of its parameters, such as amplitude, frequency, or phase,according to the information being transmitted.

Phase shift keying (“PSK”) modulation is frequently used to transmitdigital data. PSK involves shifting the phase of the carrier accordingto the value of the digital data. For example, in binary PSK (“BPSK”)the “zeros” in the digital data may be represented by a 180° shift inthe phase of the carrier, while the “ones” in the digital data may berepresented by no phase shift. Other degrees of phase shifting may beused. Quadrature PSK (“QPSK”) involves phase shifts of 0°, 90°, 180° and270°. PSK typically is referred to as “MPSK” where the “M” representsthe number of phases.

After a transmitter sends an MSPK signal over the selected transmissionmedium (e.g., telephone lines or radio frequency waves), a receiverdetects the phase changes in the accurately, the receiver must extractthe unmodulated frequency and phase (commonly referred to as thereference frequency and phase) of the carrier from the received signal.

Traditionally, phase-locked loop (“PLL”) circuits have been used toacquire carrier phase in many types of MPSK modems. PLLs are relativelyeasy to implement with either analog or digital technology and, ingeneral, are considered to have good “steady state” performance.

However, PLLs are not effective for “bursty” transmissions. That is,transmissions where the signal is received in bursts (e.g.,time-division multiple access, “TDMA,” signals), rather than as acontinuous signal. In many cases, PLLs cannot achieve fast phaseacquisition with a high probability of accuracy due to a phenomenonknown as “hang-up.” Moreover, PLLs typically have a limited frequencyacquisition range unless they are augmented with search schemes. Thesesearch schemes, however, introduce significant delay into the phaseacquisition process.

Due to the above problems and the proliferation of digital technologyand more powerful digital signal processors, many modern burst-modemodems acquire carrier phase using open-loop algorithms instead of PLLs.Open-loop solutions typically use a preamble at the beginning of eachburst. A modem that processes burst-type transmissions that include asufficiently long preamble may acquire phase using some form ofcorrelator searching for a known preamble or using a decision directedsolution. Some of these techniques are described in M. P. Fitz,“Equivocation in Nonlinear Digital Carrier Synchronizers,” IEEETransaction on Communications, vol. 39, no. 11, November 1991; and M. P.Fitz and W. C. Lindsey, “Decision-Directed Burst-Mode CarrierSynchronization Techniques,” IEEE Transactions on Communications, vol.40, no. 10, October 1992, the contents of which are hereby incorporatedherein by reference.

The preamble technique is an unsuitable solution for many applications.For example, long preambles may take up a relatively large portion ofthe burst (particularly for short bursts). This reduces the effectivebandwidth that is available for data transmission. Moreover, in someapplications there is a need to acquire phase and frequency at any pointduring the burst or to reacquire it, once it is lost. Inherently, thepreamble technique is ineffective for these applications.

Alternatively, a scheme based on a maximum likelihood algorithm may beemployed. This scheme removes the data dependency of the received signalusing a nonlinear operation. It has been shown for the case of an MPSKmodulated carrier with an unknown phase that when the frequency is known(down to a small error) the phase can be efficiently estimated using anonlinear algorithm. This technique may lead to results which are onlymoderately less accurate than those achievable by an optimal linearestimator operating on an unmodulated carrier. See, for example, thearticle by A. J. Viterbi and A. M. Viterbi entitled “NonlinearEstimation of PSK-Modulated Carrier Phase with Application to BurstDigital Transmission,” IEEE Transactions on Information Theory, vol.IT-29, no. 4, pp. 543-551, July 1983, the contents of which is herebyincorporated herein by reference.

The above techniques provide phase estimates for signals where thefrequency is known. However, many applications require frequency andphase estimation for MPSK signals with a relatively wide frequencyuncertainty range. For example, due to the instability of oscillators inthe transmitters and receivers, the frequency of the received signal maybe different than the expected frequency. Under certain circumstances,the frequency uncertainty range (i.e., range of possible frequencies ofthe received signal due to the instability) may be a significantfraction of the signal symbol rate. (In PSK, the information transferrate is defined in terms of symbols per second.) Moreover, the frequencyof the received signal typically will change over time due to theinstability. Thus, the receiver must produce continuous phase andfrequency estimates to maintain synchronization between the transmitterand receiver.

Various techniques have been proposed to determine the frequency of asignal within a known frequency uncertainty range. For example, it hasbeen shown that a maximum-posterior-probability frequency estimator mayconsist of a bank of equally spaced envelope correlation detectorsfollowed by “choose largest” logic. Viterbi, A. J., Principles ofCoherent Communications, McGraw-Hill Book Co., New York, 1966, thecontents of which is hereby incorporated herein by reference. However,this technique only dealt with an unmodulated sinusoid and did notdetect the phase of the signal.

Thus, a need exists for an efficient frequency and phase estimator forsignals that have a frequency uncertainty range that is a significantfraction of the symbol rate. Moreover, the estimator needs to produceestimates for each symbol following the initial acquisition of thesignal and do so with high probability and within a relatively smallnumber of symbols.

In accordance with a preferred embodiment of the invention, a frequencyand phase estimator divides the frequency uncertainty range of thesignal into a plurality of narrower frequency bands, the width of whichis dictated by the required frequency resolution. For example, if thefrequency uncertainty range covers 10 kHz, one band could cover thefirst 1 kHz in the range, another band could cover the second 1 kHz, andso forth. The estimator processes the signal and generates a frequencyestimate by determining the band into which the incoming signal falls.The estimator then calculates a phase estimate.

The estimator shifts the frequency of, filters and samples the incomingsignal, to produce a continuous sequence of discrete-time signal samplesfor each band. The frequency shift operation involves shifting thefrequency of the incoming signal by an amount determined by the centerfrequency of each band relative to the center frequency of theuncertainty range. For example, when there are ten bands defined, theincoming signal is frequency shifted by a different amount for each bandresulting in ten different shifts. Depending on the implementation, theincoming signal may be frequency shifted either before or after thesignal is converted to a digital format by analog-to-digital conversion.Preferably, a pair of analog-to-digital converters is utilized. Eachsymbol in the incoming signal is sampled one or more times to producethe sequence of samples.

Next, the estimator removes the PSK modulation and accumulates thesamples for each band. The modulation is removed by processing thesamples with a nonlinear algorithm. A complex accumulator (the samplesare complex numbers, i.e., vectors) then accumulates a predefined numberof the demodulated samples. Typically, each of the accumulatorsprocesses samples corresponding to same incoming symbols.

To determine which band contains the actual frequency of the incomingsignal, the estimator compares the magnitudes of the accumulatedvectors. In general, the band with the largest accumulated vector is theone associated with the incoming frequency. Thus, the estimate of thesignal frequency may be derived from the center frequency of the band.

The estimator calculates the reference phase of the received signal fromthe phase of the largest accumulated vector. Typically, this phase isadjusted to compensate for an anomaly known as equivocation.

In one embodiment, many of the above operations are implemented in adigital signal processor (“DSP”). In this case, provided the DSP hassufficient processing power, the processing operations for each band maybe accomplished in series, i.e., one band at a time. Hence, theinvention may be practiced using only a single DSP.

Thus, a system constructed according to the invention provides anefficient method of calculating the frequency error and the currentphase of a MPSK modulated signal that has a relatively large frequencyuncertainty range. As desired, the system produces a continuous streamof frequency and phase estimates. Moreover, the system produces goodestimates after processing a relatively small number of symbols.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features of the invention will become apparent from thefollowing description and claims, considered in view of the accompanyingdrawings, wherein similar references characters refer to similarelements throughout, and in which:

FIG. 1 is a functional block diagram illustrating one embodiment of afrequency and phase estimator embodying the present invention;

FIG. 2 is a flowchart illustrating operations that are performed by theapparatus of FIG. 1;

FIG. 3 is a schematic diagram illustrating a preferred embodiment of afrequency shifter that may be used in the embodiment of FIG. 1;

FIG. 4 is a functional block diagram illustrating one embodiment of asignal receiver embodying the invention, the receiver including adigital signal processor constructed;

FIG. 5 is a flowchart illustrating operations that are performed by thedevice of FIG. 4;

FIG. 6 is a block diagram illustrating another frequency and phaseestimator embodying the invention; and

FIG. 7 is a graphic illustration of the relationship between thestandard deviation of the phase estimator error and signal-to-noiseratio for BPSK, QPSK, and 8PSK, as well as the Cramer-Rao lower bound.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

In FIG. 1, a frequency and phase estimator E processes a modulatedsignal r(t) (left) to generate continuous streams of frequency estimatesand phase estimates (right). These estimates are used by a receiver (notshown) to recover information from the incoming signal. In accordancewith the invention, the frequency uncertainty range of the signal isdivided into several bands (e.g., band 1 20A through band 2k+1 20B) toaccommodate signals that may have a relatively wide frequencyuncertainty range.

The estimator E generates discrete-time samples for each of these bandsand processes the samples to provide the estimates. Initially, a downconverter 22 and a frequency rotator 24 frequency shift the incomingsignal to provide signals for each band. To produce the discrete-timesequence of samples, matched filters 26 and symbol samplers 28 filterand sample each symbol within the rotated signals. Next, a nonlinearprocessor 30 applies a nonlinear algorithm to the samples to remove themodulation from the samples. A predefined number of the demodulatedsamples are accumulated in a complex accumulator 32. The complexaccumulator 32 that contains the largest accumulated vector after aselected number of samples are accumulated identifies the band closest(in relative terms) to the frequency of the incoming signal. Thus, alongest vector selector 34 compares the accumulated vectors andidentifies the associated band (e.g., band “m”). The estimator E thencalculates the frequency estimate according to the frequency associatedwith this band. In addition, the reference phase of the signal isobtained from the argument of the accumulated vector for that band.

With the above overview in mind, FIG. 2 describes an exemplary frequencyand phase process performed by the system of FIG. 1, beginning at block200. At block 202 the baseband down converter 22 converts the modulatedcarrier signal r(t) to baseband.

To generate the estimates, the incoming signal is processed over aperiod which spans N consecutive symbols: n=n₀, . . . , n₀+N−1. Duringthe n^(th) symbol time the received signal r(t) may be represented as:$\begin{matrix}{{{{r(t)} = {{\sqrt{\frac{2E_{s}}{T}}{{Cos}\left( {{\left( {\omega_{c} + {\Delta\omega}} \right)t} + \theta + \theta_{n}} \right)}} + {n(t)}}};}{{\left( {n - 0.5} \right)T} < t < {\left( {n + 0.5} \right)T}}} & (1)\end{matrix}$

In Equation 1, θ_(n), which is the information carrying parameter, maychange from symbol to symbol subject to the following constraint:$\begin{matrix}{{\theta_{n} = {i_{n}\left( \frac{2\Pi}{M} \right)}};{i_{n} = {{0.1\quad \ldots \quad M} - 1}}} & (2)\end{matrix}$

where θ is a fixed, unknown phase. T is the duration of a symbol. E_(s)is the energy of the signal per symbol.

The parameter n(t) is white Gaussian noise with one-sided power spectraldensity N₀. This means that the covariance function of the noise is:

R(τ)=N ₀/2δ(τ)  (3)

where δ ( ) is the Dirac delta function.

The signal to noise ratio (“SNR”) of a signal is defined as:SNR=2E_(s)/N₀. This is the ratio between the peak instantaneous signalpower and the mean noise power at the output of a filter matched to thesignal in Equation 1 at the sampling instance. The term E_(b)/N₀ (a termcommonly used in the art) is then:

E _(b) /N ₀=(E_(s) /N ₀)/Log₂ M=(SNR/2)/Log₂ M  (4)

Typically, the precise frequency of the signal ω_(c) is not known dueto, for example, the instability of the oscillators in the transmitterand receiver. Therefore, in accordance with the invention, anuncertainty region for ω_(c) is defined as a band of width W centeredaround ω_(c) (block 204). The band W is divided into 2k+1 equal bandscentered around ω_(c): $\begin{matrix}{{\omega_{c} - {k\frac{\Omega}{{2k} + 1}}},\ldots \quad,\omega_{c},\ldots \quad,{\omega_{c} + {k\frac{\Omega}{{2k} + 1}}}} & (5)\end{matrix}$

At block 206, the frequency rotator 24 (e.g., a down converter)associated with each band shifts the frequency of the in-phase (“I”) andquadrature-phase (“Q”) symbols produced by the baseband down converter22. Each frequency rotator 24 in the bank of 2k+1 staggered frequencyrotators 24 is tuned to the center frequency of one of the bands.

FIG. 3 is a schematic of a simplified circuit that rotates a signal witha frequency of Ω radians per second by “ω” radians per second.Multipliers 42 generate products of the incoming signals that are summedby adders 44 to produce a signal with the desired frequency (i.e., Ω+ω).Many variation of this technique are possible, some of which arediscussed below. In addition, it would be apparent to one skilled in theart that the operations of the baseband converter 22 may be combinedwith the operations of the frequency rotators 24, if desired.

Referring again to FIG. 2, at block 208 the outputs of each rotator 24are filtered by a pair of matched filters 26. Assuming a “square”unfiltered symbol shape, a good choice for the matched filter 26 is anintegrator.

The filtered output is sampled to generate the discrete-time sequence ofsamples (block 210). For example, the integrator (not shown) is reset att=(n−0.5)T_(s). The symbol sampler 28 samples the output of theintegrator at t=(n+0.5)T_(s). The n^(th) sample of the in-phase i^(th)frequency rotator 24, which is tuned to: $\begin{matrix}{\omega + {i\frac{\Omega}{{2k} + 1}}} & (6)\end{matrix}$

is: $\begin{matrix}{{I_{i}(n)} = {{\sqrt{E_{s}\frac{T}{2}}{{Cos}\left( {{\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right){nT}} + \theta + \theta_{n}} \right)}{Sinc}\frac{T}{2}\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right)} + {n_{I,i}(n)}}} & (7)\end{matrix}$

where Sinc(x)ΔSin(x)/x and the quadrature sample is:

$\begin{matrix}{{Q_{1}(n)} = {{\sqrt{E_{s}\frac{T}{2}}{{Sin}\left( {{\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right){nT}} + \theta + \theta_{n}} \right)}{Sinc}\frac{T}{2}\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right)} + {n_{Q,i}(n)}}} & (8)\end{matrix}$

where n_(I,i)(n)=n_(Q,i)(n) are independent Gaussian random variableswith σ² _(I,i)(n)=σ² _(Q,i)(n)=N₀T/4. i=−k, . . . , k.

Normalizing the peak signal power and the noise variance by dividingboth by E_(s)T/2 (which is the maximum squared signal peak amplitude),the peak amplitude of the signal is then “1” and the variance of thenoise components is: $\begin{matrix}{\sigma_{1}^{2} = {\sigma_{Q}^{2} = {{\frac{N_{0}T}{4}\frac{2}{E_{s}T}} = \frac{N_{0}}{2E_{s}}}}} & (9)\end{matrix}$

Hence: $\begin{matrix}{\sigma_{i} = {\sigma_{Q} = {\sqrt{\frac{N_{0}}{2E_{s}}} = \frac{1}{\sqrt{SNR}}}}} & (10)\end{matrix}$

The noise samples, taken simultaneously at the end of every symbol, arein general correlated. The elements of the covariance matrix of thenoise samples of the 2k+1 channels may be derived through knownprocedures. The covariance is a fixed (i.e., independent of n) 2k+1 by2k+1 matrix. It is normalized by multiplication by 2/(E_(s)/T).

At block 212, if each symbol is to be sampled more than once, the aboveprocess is repeated (with some modification). The dashed line from block212 represents one possible multi-sampling method. Multi-sampling isdiscussed in more detail below.

After the estimator E generates the discrete-time samples, a nonlineardemodulator 30 removes the data dependency of the samples (block 214).Defining Φ(n) as the argument (i.e., angle) of the vectorX_(i)(n)=I_(i)(n)+jQ_(i)(n). In the absence of noise: $\begin{matrix}\begin{matrix}{{\phi_{i}(n)} = \quad {\arg \left\{ {X_{i}(n)} \right\}}} \\{= \quad {{Tan}^{- 1}\left( {{Tan}\left\lbrack {{\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right){nT}} + \theta + \theta_{n}} \right\rbrack} \right.}} \\{= \quad {{\left( {{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} \right){nT}} + \theta + \theta_{n}}}\end{matrix} & (11)\end{matrix}$

It is apparent that Φ(n) is dependent on θ_(n). As mentioned above,θ_(n) is the phase shift due to the information that modulated thecarrier. To retrieve the reference phase of the carrier, the effects ofθ_(n) are eliminated.

To eliminate the unknown θ_(n), the nonlinear demodulator calculatesZ_(i)(n)=F{¦X_(i)(n)¦}Exp{jψ_(i)(n)}, where the nonlinear function F{ }is discussed below and:

ψ_(i)(n)=M _(Φi)(n)  (12)

Since Mθ_(n) is an integer multiple of 2π, Z_(i)(n) is independent ofθ_(n). Moreover, it is not necessary to preserve the correct quadrantwhen calculating Φ_(i)(n) because practical values of M are even. Forexample, when X_(i)(n) and Q_(i)(n) are both negative, Φ_(i)(n) could bechosen in the first quadrant, etc.

Regarding, the choice for the function F{ }, functions of the form:F{x}=x^(α) have been studied for a phase estimator where the frequencyis assumed known or where the frequency error is known to be very small.See the Viterbi article referenced above and B. E. Paden, “A MatchedNonlinearity for Phase Estimation of a PSK-Modulated Carrier,” IEEETransactions on Information Theory, vol. IT-32, no. 3, pp. 419-422, May1986, the contents of which is hereby incorporated herein by reference.The conclusion derived from some theoretical analysis and simulationsfor M=4 is that for E_(b)/N₀>6 dB, α=1, or in other words, F{x}=x may bepreferred. For E_(b)/N₀<6 dB, α=2 may be better, and for asymptoticallylow values of E_(b)/N₀, α=4 appears to be preferred. The variable αplays an additional role because it modifies the effect of the Sinc( )terms in EquationS 7 and 8. When α>0, vectors accumulated by channelsfurther away from the correct one have that part of their magnitude(which is related to the signal) diminished. This phenomenon ismeaningful only for large values of ΩT. Moreover, the gain that can beobtained from using α>0 is small and, in practical situations whenE_(b)/N₀>0 dB and ΩT<π/2, may not warrant the additional processingload.

At block 216, the complex accumulator 32 calculates the magnitude ofvector Z(n) and adds it to the accumulator 32. This process starts withthe n₀ ^(th) symbol and continues for N consecutive symbols (block 218).The best estimate of the phase may be achieved for the center samplewhen N is odd. See the Viterbi article referenced above and D. C. Rifeand R. R. Boorstyn, “Single-Tone Parameter Estimation from Discrete-TimeObservations,” IEEE Transactions on Information Theory. vol. IT-20, no.5, pp. 591-598, September 1974, the contents of which is herebyincorporated herein by reference. Thus, n₀ is selected as: n₀=−(N−1)/2.As shown in FIG. 1, 2k+1 vectors Z_(i)(n), i=−k, . . . , 0, . . . , k,are calculated (one for each band) and added to the correspondingaccumulator 32.

Blocks 206 to 218 in FIG. 2 describe operations that may be performedfor each of the bands (e.g., band 1 20A) in FIG. 1. In the embodiment ofFIG. 1, these operations typically would be performed in parallel.However, they could be performed one band (or a few bands) at a time.One example of serial processing is discussed below.

When N vectors have been added to all the accumulators 32, theaccumulator 32 holding the longest vector (largest in absolute value) isselected (block 220). In other words, let $\begin{matrix}{{{Z_{i,T}\overset{\Delta}{=}{\sum\limits_{n = {{- {({N - 1})}}/2}}^{{({N - 1})}/2}{Z_{i}(n)}}};\quad {i = k}},\ldots \quad,0,{\ldots \quad k}} & (13)\end{matrix}$

be the final content of accumulator i. Zm,T is then the largest vector:

Z _(m,T)=max_(i) {Z _(i,T)}  (14)

At block 222, a frequency estimate calculator 38 generates Δω from theindex of the selected band:

Δω_(est) =m{Ω/(2k+1)}  (15)

At block 224, a phase estimate calculator 40 generates θ from theargument of the longest vector:

θ_(est)=argument{Z _(m,T) }/M  (16)

If argument {Z_(m,T)}/M spans a range −π to π, the phase estimate willbe confined to a range −π/M to π/M. Although the actual phase of thetransmitter progresses (if Δω≠0) and may drift in the entire band ofwidth 2π, the string of estimates will be broken into segments that spanone sector only. The source of this problem is the multiplication of thephase Φ by M to yield ψ (which is interpreted by the algorithm as ψModulo 2π) followed by the division by M in Equation 16. Moreover, thealgorithm suffers from so-called “equivocation,” an anomaly describedand analyzed in the 1991 article by Fitz referenced above.

At block 226, an estimate unwrapper 36 handles this problem as follows.θ(j) is defined as the argument of Z_(m,T)(j), i.e., the argument of thelongest vector calculated by the j^(th) application of the algorithm.Then, θ_(est)(1)=θ(1)/M (recall that the first phase estimate isarbitrary anyway). Next, θ_(p)(j+1) is defined as:

Θ_(p)(j+1)=Θ(j)+Δω_(est)(j)T; j=1,2,3, . . .  (17)

θ_(p)(j+1) is the predicted value of θ(j+1) based on θ(j) and thefrequency estimate performed during the j^(th) application of thealgorithm. θ_(a)(j+1) is defined as:

Θ_(a)(j+1)=Θ(j+1)+k2π  (18)

where k is an integer defined by:

|Θ(j+1)+k2π−Θ_(p)(j+1)|≦|Θ(j+1)+i2π−Θ_(p)(j+1)|  (19)

for any integer i≠k. Then:

Θ(j+1)=Θ_(a)(j+1)  (20)

and $\begin{matrix}{{\Theta_{est}\left( {j + 1} \right)} = {{MOD}_{2\pi}\left\lbrack {{\Theta_{est}(j)} + \frac{{\Theta \left( {j + 1} \right)} - {\Theta (j)}}{M}} \right\rbrack}} & (21)\end{matrix}$

With the above definition, θ_(p)(j) and θ(j) can have any values. Toprevent indefinitely large (or small) values, modulo M2π values may beused instead.

The above technique generates a “continuous” sequence of unwrapped phaseestimates as long as ¦θ(j+1)−θ_(p)(j+1)¦<π. Whenever this condition isviolated due to excessive noise, the reconstructed sequence θ_(est) islikely to jump to a different sector. It then continues normal operationuntil another error causes a second jump. At sufficiently high SNRs,these jumps are infrequent.

Even if the tracking is done perfectly, however, the ambiguity remainsbecause the initial decision of where to place θ_(est) (1) is arbitrary.However, this is an ambiguity that it typically present in coherent MPSKreceivers. There are known ways to deal with this problem. See, forexample, the Viterbi article referenced above.

As in any algorithm that processes sampled data, the above algorithm issubject to aliasing. For example, assuming the noise is negligibly smalland the unknown frequency error is precisely on channel i, i.e.:$\begin{matrix}{{{\Delta\omega} - {i\frac{\Omega}{{2k} + 1}}} = 0} & (22)\end{matrix}$

For any j such that: $\begin{matrix}{{{{\left\lbrack {{\Delta\omega} - {j\frac{\Omega}{{2k} + 1}}} \right\rbrack T} = {12\frac{\pi}{M}}};\quad {l = {\ldots - 2}}},{- 1},1,2,\ldots} & (23)\end{matrix}$

the phase of the vector X_(j)(n)=I_(j)(n)+jQ_(j)(n) differs fromX_(i)(n) by an integer multiple of (2π)/M. When the algorithm multipliesthe phase by M, the resultant phase is indistinguishable fromarg{X_(i)(n)}. Therefore, all the vectors produced by those channels addin-phase, the same as those of channel i. Only the Sinc( ) term, whichaffects the amplitude of the accumulated vectors (for α≠0), and thenoise determine which accumulator ends up the largest.

The algorithm, therefore, tends to generate multiple peaks (asrepresented by a graph of the absolute value of the final contents ofthe 2k+1 accumulators). The above may happen when: $\begin{matrix}{{{{i - j}}\Omega \quad \frac{T}{{2k} + 1}} \geq \frac{2\pi}{M}} & (24)\end{matrix}$

or, since |i−j|≦2k, if $\begin{matrix}{{\Omega \quad T} \geq {{\frac{2\pi}{M}2k} + \frac{1}{2k}} \simeq \frac{2\pi}{M}} & (25)\end{matrix}$

Thus, aliasing may occur when ΩT>(2π/M). The Sinc( ) term has only asmall effect when ΩT=(2π/M) and M≧4. For example, in one test with M=4and α=2, the magnitude of the (closest) false peak falls byapproximately 2 dB in comparison with the correct peak.

One solution for the situation when Ω>2π/M is to sample more than onceper symbol. The following example illustrates this for the M=4 case.Referring to FIG. 2, at block 208, the estimator integrates over thefirst half of each symbol. At block 210, the estimator samples theresult, then resets (dumps) the integrator and repeats the operations ofblocks 208 and 210 for the second half of the symbol. This producestwice as many samples, 2(2k+1)2N altogether, for a sequence of Nsymbols. The estimator multiplies the argument over every vector by fourbefore summing them all up as before.

By sampling twice per symbol, Ω may be twice as large as before and theestimator E still avoids aliasing. In fact, Ω can be increased by P ifthe estimator E uses P samples. However, some loss in SNR will resultfrom this approach.

The magnitude of this loss can be simulated. For the case of two samplesper symbol, the signal and the random noise components for all thesamples are calculated. First, I_(i,T/2)(n) and Q_(i,T/2)(n) , thein-phase and quadrature signals accumulated by the i^(th) channel duringthe first half of the n^(th) symbol (from t=nT−(T/2) to nT) arecalculated. Then, the samples taken at the end of the symbol (I_(i,T)(n)and Q_(i,T)(n)) are calculated (from t=nt to nT+(T/2)). Similarly, thenoise samples (n_(I,i,T/2)(n) and n_(Q,i,T/2)(n)) taken at the middle ofthe n^(th) symbol (from t=nT−(T/2) to nT) are calculated as are thosetaken at the end of the n^(th) symbol (from t=nt to nT+(T/2)).

Comparing the frequency estimation results for the single and doublesampling case, for 0.1 radians/symbol as a criterion, a SNR loss ofapproximately 1.3 dB has been calculated. As for the phase estimationand 0.1 radian as a criterion, a loss of approximately 1 dB has beencalculated.

Referring to FIG. 4, an alternative embodiment of the invention thatuses a digital signal processor (“DSP”) 46 is shown. FIG. 4 also depictsa typical implementation where the estimator is incorporated into areceiver R. The receiver R includes a signal decoder 48 that uses thefrequency and phase estimates to decode the modulated information fromthe incoming signal.

Referring briefly to FIG. 1, it may be seen that except for a common“front-end” and a common “back-end” the estimator includes 2k+1“channels” (i.e., bands) which differ only in the amount (and sign) of“rotation” that they perform in front of the matched filters 26 (e.g.,integrators). Depending on the actual set of parameters (e.g., thesymbol rate) and availability of a fast DSP device, it may be possibleto perform the calculations for all 2k+1 channels serially in thedigital domain, excluding possibly the first (common) down-converter 22.

An exemplary operation of the embodiment of FIG. 4 is treated in FIG. 5beginning at block 250. At block 252, the incoming signal is downconverted to baseband as discussed above in conjunction with FIG. 1.

A dual analog to digital converter (“ADC”) 50 at the quadrature outputsof the common down converter 22 converts the analog signals to digitaldata streams (block 254). That is, the ADC 50 samples the in-phase andquadrature symbols. As discussed above, each symbol may be sampledmultiple times (e.g., “x” times per symbol). These samples may be usedby the DSP 46 in “real-time” or stored in a memory 51 to be used asneeded by the DSP 46.

Next, the estimator E selects a channel to process (block 256).Initially, the estimator E will process signals for each of the channels(as discussed below). However, in many practical situations, it is notnecessary to continue to perform all the calculations in real time. Whenthe frequency is known to be relatively stable, it is possible toestimate the frequency even when the actual calculations last many timesthe duration of N symbols. This operation must be repeated from time totime, of course, to track frequency drifts. Once the frequency is known,the phase can be derived by activating only one channel, the one thatcorresponds to the correct frequency.

At block 258, to provide the desired frequency rotation, a samplegenerator 52 produces samples of the sine and cosine functions for eachchannel. A different set of samples will be generated for each channel.At block 260, a multiplier 54 multiplies samples of the incoming signalby the samples of the sine and cosine for the selected channel.

In the embodiment of FIG. 4, the filters are implemented in the digitaldomain (block 262). In some implementations, these filters may employ afilter design other than a simple integrator. For example, finiteimpulse response (“FIR”) and infinite impulse response (“IIR”) filters.In practice, the rotation operation may be considered part of thefiltering operation. As mentioned above, depending on the availablecomputing power, the rotating, filtering, nonlinear processor(demodulator) 58 (block 264) and accumulator 60 (block 266) operationscan be performed serially. In general, these basic operations asperformed by the DSP 46 are similar to those discussed above inconjunction with FIG. 1.

After the above operations are completed for each channel (block 268),the choose largest logic 62 (e.g., choose largest vector) calculates thelargest vector (block 270). As discussed above in conjunction with block256, a channel activator 64 may store the identity of the selectedchannel (e.g., “m”) and control the selection of the channel in futurephase estimation operations. In this case, once the channel has beenselected, the operation of blocks 268 and 270 may be omitted until thefrequency estimate is recalculated.

The basic operations of the remaining steps performed by the DSP aresimilar to those discussed above in conjunction with FIG. 1. Thus, aphase estimator 66 calculates a phase estimate (block 272). A phaseunwrapper 68 processes this estimate to generate an unwrapped phaseestimate (block 274). A frequency estimator 70 generates a frequencyestimate (block 276). At block 278, the latter two estimates are sent tothe signal decoder as discussed above.

The DSP embodiment of FIGS. 4 and 5 thus provides an attractive methodof practicing the invention. In particular, it may be implemented usingonly one down converter, thereby possibly reducing the cost of thesystem.

Several aspects of the operation of the embodiments discussed aboveshould be noted. In general, the frequency estimate is biased. Theprobability density function of Δω_(est) is symmetric around Δω onlywhen Δω=0. Recall that there are exactly 2k+1 discrete frequencyoutcomes. When Δω is not equal to any possible outcome, an error mustoccur and the outcome is biased toward the closest possible outcome.When Ω/k is small enough so that this phenomenon can be ignored, thealgorithm is still biased when Δω≠0, i.e., when Δω is not at the centerof the frequency uncertainty range. When Δω is closer to one end of thefrequency uncertainty range, the estimate is biased toward the otherend. This last effect diminishes as the values of the SNRs increase, andincreases when Δω is very close to the band edge. A similar situationhas been reported in the Rife and Boorstyn article referenced above.

The performance of the device may depend on the frequency off-setbetween the received signal and the “closest” channel. If i is the indexof the closest channel, then [Δω−iΩ/(2k+1)]T is the phase shift betweensuccessive vectors accumulated by channel i. Thus, given a frequencyresolution of ΩT/(2k+1) radians/sample, there is a maximum value of N(e.g., N_(m)) beyond which the performance will start dropping. Withasymptotically low noise N_(m) corresponds to:

N _(m) [Δω−iΩ/(2k+1)]T≈π  (26)

For practical values of SNR, N should be chosen lower than that.

In general, the accuracy of the frequency estimate depends on theresolution of the bands. That is, the narrower the band, the moreaccurate the frequency estimate. This accuracy comes at the expense,however, of added cost (e.g., more DSP operations per received symbol).

FIG. 6 illustrates the performance of various embodiments of theinvention in comparison with the Cramer-Rao lower bound. Specifically,this figure compares graphically the standard deviation of the phaseestimate as a function of signal-to-noise ratio, 2E_(s)/N₀. on thevariance of the estimates follows. These results were obtained bycomputer simulation using MATLAB. To bypass the difficult task ofcalculating the bound, the approach described in the Viterbi article ofmaking comparisons with the known bound for the M=1 case may be adopted.More specifically, the simulated results obtained for the variance ofthe phase estimation error using this algorithm are compared with theCramer-Rao lower bound for a single sinusoid with known frequency,duration NT and energy NE_(s). For this particular case, the bound is:$\begin{matrix}{{{var}\left\{ \theta_{est} \right\}} \geq \frac{1}{N\left( {2{E_{s}/N_{0}}} \right)}} & (27)\end{matrix}$

The simulations of FIG. 6 utilized these operational parameters: k=40;α=1; ΩT=2.025 radians/symbol; one sample per symbol; and ΔωT isuniformly distributed in the range 0 to 0.025 radians/symbol. FIG. 7shows that the signal-to-noise threshold of the M=8 curve is higher thanthat of M=4, while M=2 has the lowest threshold. Above the threshold,the estimator approaches the bound. When the signal-to-noise ratiodecreases below the threshold, σ_(θ) climbs and saturates at a level forM=8 which is higher than that for M=4 and highest for M=2. When thesignal-to-noise ratio approaches zero, the distribution of the phaseestimates produced by the algorithm approaches a uniform distributionover the sector −π/M to π/M, and therefore the phase estimate varianceapproaches (π/M)²/3, which is a function of M.

Simulations of the invention were accomplished as follows. For each run,a signal, consisting of N random symbols, may be generated. Then, foreach symbol, 2N(2k+1) noise samples (4N(2k+1) for the two symbols persample case) may be generated. These noise samples represent the noisecomponents appearing at the output of the 2(2k+1) matched filters (seeFIG. 1) at each sampling instance. The noise samples are allstatistically independent for different sampling times, but should bemutually chosen correctly for every one sampling time so as to match thecovariance matrix, which is a function of n.

Y is defined as a random row vector containing 2(2k+1) components(4(k+1) components for the two samples per second case), where eachcomponent is a statistically independent, identically distributedGaussian random variable with zero mean and variance equal to one. [R]is defined as the required covariance matrix. The linear transformation:X=Y[R]^(½) generates a row random vector X such that E{X′X}=[R], whereX′ is the transpose of X.

As noted above, the Sinc( ) terms in EquationS 7 and 8 departsignificantly from 1 only for values of ΩT exceeding π/2 radian/symbol.When ΩT is smaller than π/2, the bank of matched filters may be replacedwith two filters, one for the I channel and one for the Q channel. Thisconfiguration is depicted in FIG. 6. The bank of rotators 72 are placedat the output of the filters 74.

The structure and method taught by the invention may also be used toestimate the frequency and phase of a differentially MPSK modulatedcarrier. In addition, with minor modifications, the teachings of theinvention may be used for modulation schemes such as IS-136, where thephase of each successive symbol is incremented at the transmitter by afixed known amount, independent of the phase shifts attributable to themodulating data.

The embodiments described above illustrate that the invention may bepracticed in a wide variety of configurations and the functionsdescribed above may be distributed among various components. Forexample, the functions for each band (channel) may be incorporated intoone or more devices. The system may be expanded to accommodate differentuncertainty ranges and different degrees of resolution for the bands. Abank of DSPs may be used to process the channels in parallel.

Typically, the DSP operations would be implemented as software routinesinstalled on and executed by a DSP device such as a “DSP-2000” availablefrom Lucent Technologies. Alternatively, one or more of the aboveoperations could be implemented in another hardware device such as amicroprocessor, a custom integrated circuit, etc. These designselections would depend on the requirements of the specificimplementation.

From the above, it may be seen that the invention provides an effectivefrequency and phase estimator that provides a number of advantages overconventional systems. For example, the estimator automatically adjuststo changes in the frequency of the incoming signal. Continuous estimatesare provided. No preamble is needed. Read-only-memories are not employedfor the nonlinear algorithm.

While certain specific embodiments of the invention are disclosed astypical, the invention is not limited to these particular forms, butrather is applicable broadly to all such variations as fall within thescope of the appended claims. To those skilled in the art to which theinvention pertains many modifications and adaptations will occur. Forexample, various methods of down converting and frequency rotating maybe used in practicing the invention. A variety of methods may be usedfor the sampling, filtering and accumulating operations. A number ofnonlinear methods may be used to remove the modulation. Similarly,various frequency calculating, phase calculating and unwrappingalgorithms may be utilized. Thus, the specific structures and methodsdiscussed in detail above are merely illustrative of a few specificembodiments of the invention.

What is claimed is:
 1. A method for estimating the frequency and phaseof a PSK modulated signal having a frequency uncertainty range, saidmethod comprising the steps of: frequency shifting said signal to afrequency band within said frequency uncertainty range; performing anonlinear operation on said frequency shifted signal to demodulate saidfrequency shifted signal; generating a frequency estimate of said signalaccording to said frequency shifted signal; and generating a phaseestimate of said signal according to said demodulated signal.
 2. Themethod according to claim 1 further including the step of frequencyshifting said PSK modulated signal to at least one additional frequencyband within said frequency uncertainty range.
 3. The method according toclaim 2 further including the steps of: accumulating a plurality ofsignal samples of the frequency shifted signal in each of said bands;and selecting the frequency estimate in one of said bands, based uponthe result of said accumulating step.
 4. The method according to claim 3wherein said frequency estimate comprises a frequency deviation(Δω_(est)) defined by: Δω_(est)=(mΩ)/(2k+1) where m is an indexassociated with said one selected band, Ω defines said uncertainty rangeand 2k+1 defines a number of said bands.
 5. The method according toclaim 3 wherein the accumulating step produces a vector corresponding toeach band and the selecting step selects that band corresponding to thelongest vector.
 6. The method according to claim 5 wherein said phaseestimate (θ_(est)) is determined as: θ_(est)=arg(z _(m,T))/M wherearg(z_(m,T)) is the argument of the longest vector and M is the numberof phase possibilities in the PSK modulated signal.
 7. The methodaccording to claim 1 further including the step of down-converting saiddemodulated signal to a baseband signal.
 8. The method according toclaim 1 further including the step of unwrapping said phase estimate. 9.The method according to claim 1 wherein a single frequency band isselected for estimating said phase.
 10. The method according to claim 1further including the step of taking a plurality of samples of a symbolof said demodulated signal.
 11. The method according to claim 1 furtherincluding the steps of: performing an analog-to-digital conversion onsaid demodulated signal to generate a digital sample of said signal;generating a sequence of samples representing the frequency associatedwith said band; and multiplying said digital sample with said samplewith said frequency representing samples.
 12. In a system for estimatingthe frequency and phase of a modulated input signal having a frequencyuncertainty range, the combination of: a frequency shifter for frequencyshifting said input signal to a frequency band within said frequencyuncertainty range; a nonlinear processor responsive to said frequencyshifted signal to remove modulation therefrom, creating a demodulatedsignal; a frequency estimator responsive to said frequency shiftedsignal and generating a frequency estimate of said input signal; and aphase estimator responsive to said demodulated signal and generating aphase estimate of said input signal.
 13. The combination according toclaim 12 further comprising at least one down-converter for convertingsaid input signal to a baseband signal.
 14. The combination according toclaim 12 further comprising at least one additional frequency shifterfor frequency shifting said input signal to at least one other frequencyband within said frequency uncertainty range.